impedance matching

In high frequencycircuits, the wavelength of the signals in the
circuit becomes as small as or smaller than the size of the physical
components that make up the circuit. This can cause interesting
effects that are completely non-intuitive.

In particular, two electrical components that are connected and
pass a signal between them are required to have the same impedance
to prevent reflections of the signal from occurring. Reflections due
to un-matched impedances can carry significant power; enough to damage
sensitive components that may be part of the circuit. The reflections
can also seriously degrade a signal.

Each of these techniques works in a relatively narrow band. They are
generally designed around a single frequency that you want to carry,
as the wavelength determines the length of the quarter wavelength
transformer, and the length of the stubs for the stub tuners.

As a design problem, impedance matching is all about choosing the source and load impedances to make the circuit behave well for its intended purpose. To ground our discussion in reality, let's consider the common example of an audio amplifier and a loudspeaker.

In this circuit, the generator is the amplifier output, which is an energy source. What it produces is an audio signal, which is a mix of many frequencies. However, when analysing circuits, we usually begin by considering behaviour at one frequency - otherwise it's just too complex.

The source and load impedances are each a measure of how much those elements impede the flow of energy. Impedance is analogous to resistance, but depends on frequency. It's worth reviewing these terms briefly.

Resistance is somewhat akin to friction in an electrical circuit. It is a measure of how much a circuit element resists current flow. It does not vary with the frequency of the electrical signal.

An impedance can either be capacitative (caused by a capacitor, and infinite at zero frequency, and tending to zero as frequency approaches infinity) or inductive (due to an inductor, and behaving conversely). Note, it can be one or the other, but not both at our chosen frequency. (Impedances can, and do, change from being capacitive at some frequency to inductive at some other - between the two there is a point of balance called resonance. At that frequency - the resonant frequency - they are purely resistive. But I digress.)

In fact, at a chosen frequency (I repeat this because it is important), real (what we call complex) impedances are always some mixture of resistance and (either capacitive or inductive) impedance. The whole lot can be boiled down to some vector quantity at that frequency. (It is a vector because there is a component of magnitude and also a phase angle. For this topic, we will pretend the phase angle does not exist, and just treat the impedance as if it is a simple number. We can get away with it, for the purpose of this discussion.)

Back to our circuit. The amplifier output has impedance, predominantly resistive. This is called the internal impedance of the output.

(There is also the wiring, which is almost completely resistive in nature - despite what proponents of magic audio cable might say. It is very small, unless you're using cheap bell wire or something. We just pretend that it is perfect here.)

The speaker is the load. Its resistance is usually 8, maybe 4, occasionally 2 ohms. But loudspeakers, being coils, are inductive. Not only that, but being the electro-mechanical beasts that they are, they are very complex, impedance wise - due to all the pushing and pulling that goes on, to get the cone moving and the air vibrating.

As a result, the impedance of loudspeakers varies radically, perhaps by a factor of 100 or more, across the audio frequency range. It gets more or less larger at high frequencies (it can never be less than the 8 ohms resistance), but if you see impedance as a graph against frequency, it looks like a mountain range.

For a given fixed source impedance, we get two effects as we increase the load:
- total current in the circuit decreases.
- a greater proportion of the source voltage appears across the load, and a smaller proportion is across the source.

The electrical power in a load is defined as the product of voltage and current, and if we want to maximise it, it turns out that the way to do that it to make source resistance (Rs) and load resistance (Rl) equal.

(In this type of discussion, it's only a matter of time before we start calling these quantities by letters.)

However, that has an unfortunate side effect (in the case of our real hifi amp). When source and load are equal, the power dissipated in the source is also maximum.

This really is bad news. Amplifiers have all kinds of stuff (heatsinks, fans ...) in them to get rid of the heat they dissipate, which of course is a result of the power they burn off internally. Heat kills transistors. We don't want lots of power to be consumed in the source impedance.

A second very bad side effect appears if we make Rl = Rs (at our chosen frequency). At other frequencies, because of the large variation of load impedance with frequency, the power dissipated in the load, and therefore the perceived volume, will vary hugely.

So, the result of designing for maximum power transfer condition would be a very hot, possibly burned out, amplifier, and a very bad sounding system. (These exist, but they aren't intentional.)

In practice, we make the amplifier output impedance as small as possible, at all frequencies. Now, even at high currents, the voltage across the amp, and therefore the power burned inside it, is small.

A typical figure for amp output impedance might be something like 0.1 ohms, or even less.

A second side effect of this is that even if the speaker impedance varies from its theoretical minimum (at DC, a frequency of zero) of 8 ohms to several thousand ohms at high frequency, the voltage across the speaker remains pretty much the same, and frequency response does not suffer much.

Impedance matching is fairly interesting as an electrical phenomenon because it describes how to get energy from a source to a load. If you design to maximise load power, you also do maximum work on the source side. By sacrificing a little load power you simplify the design. You gain efficiency and performance by accepting slight losses, and as a result you run cooler and sound better.

The phrase impedance matching seems to have moved towards mainstream language in recent times:

An ex-boss described the role of tech support as 'impedance matching between developers and the customer'.

More recently, I read a software engineer describe generated code as being often poor because of the 'impedance mismatch between languages'.

Even though both these individuals must have some electrical knowledge, it's interesting to see this specialised concept get used like this. Please msg me if you have any other occurences to report.